IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
Citation
Keivan Zavari, Goele Pipeleers, Jan Swevers, 2014,
Gain-scheduled controller design: illustration on an overhead crane
IEEE Transactions on Industrial Electronics, Vol. 61 , No. 7, pp. 3713 - 3718.
Archived version
Author manuscript: the content is identical to the content of the published paper, but without
the final typesetting by the publisher
Published version
http://dx.doi.org/10.1109/TIE.2013.2270213
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E-mail: keivan.zavari@kuleuven.be
Phone number: +32 16 32 92 61
IR
lirias, KULeuven link
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
2
Gain-scheduled controller design: illustration on an
overhead crane
Keivan Zavari, Goele Pipeleers, and Jan Swevers,
Abstract—This paper extends a recently developed
interpolation-based approach to design gain-scheduled
controllers for linear parameter varying systems with a thorough
evaluation, comprising both simulations and experiments, on
an overhead crane system. In the first step of the approach,
linear time-invariant controllers are designed for local working
conditions of the system using a multi-objective H∞ method.
With the help of this method, the fundamental trade-off between
reference tracking and disturbance rejection in the overhead
crane control problem is analyzed. In the second step, a statespace interpolation method is used to calculate a gain-scheduled
controller. Although this approach does not guarantee stability
and performance under parameter variations, experiments on
the crane setup show that these variations do not compromise
the performance of the obtained controller.
Index Terms—LPV systems, H∞ design, Gain-scheduling controller.
I. I NTRODUCTION
INEAR Parameter Varying (LPV) models describe linear
system dynamics that vary as a function of time-varying
parameters, called scheduling parameters. Although unknown
a priori, these parameters can generally be measured or
estimated in real time. To achieve satisfactory stability and
performance for all possible parameter trajectories, LPV models demand so-called gain-scheduling (GS) controllers, which
use the measurement/estimate of the scheduling parameters in
computing the control signal [1], [2]. In the last decade, GS
control has received significant practical interest and research
effort. Many mechatronic systems, such as overhead cranes
[3], electromagnetic actuators [4], wind energy conversion systems [5], [6] and surgical teleoperation systems [7] do exhibit
LPV dynamics. In addition, LPV models also result from
linearizing nonlinear system dynamics around a given statetrajectory. Hence, GS control is an effective and economical
method for nonlinear control design in practice.
The GS controller design approach adopted in this paper
is an interpolation based approach: First a set of local linear
time-invariant (LTI) controllers are designed for different local
working conditions of the LPV system, and subsequently,
these LTI controllers are interpolated to obtain a GS controller
as a function of the scheduling parameter. Interpolation based
L
Manuscript received October 1, 2012. Accepted for publication May 17,
2013.
c 2009 IEEE. Personal use of this material is permitted.
Copyright However, permission to use this material for any other purposes must be
obtained from the IEEE by sending a request to pubs-permissions@ieee.org.
K. Zavari, G. Pipeleers and J. Swevers are with the Department
of
Mechanical
Engineering,
KU
Leuven,
Belgium,
e-mail:
keivan.zavari@mech.kuleuven.be.
methods are generally considered practical and intuitive although contrary to the LPV GS controller design approaches
(e.g. [8], [9], [10]) they cannot provide a guarantee for stability
and performance.
This paper complements the work presented at the 12th
International Workshop on Advanced Motion Control in Sarajevo, Bosnia [11]. As it is described in [11], our GS controller
design approach uses a multi-H∞ method to design local LTI
controllers, and interpolates them into a GS controller using a
recently developed LPV modeling method called state-space
model interpolation of local estimates (SMILE) [12]. The
present paper extends that work by a thorough validation of the
approach on an overhead crane setup, which is a mechanical
system with a lightly damped resonance.
The overhead crane system exhibits LPV dynamics where
the cable length is the scheduling parameter. Although GS
control has been applied to overhead cranes (e.g. [3], [13],
[14]), these controller designs mainly focus on swing angle
disturbance rejection and do not investigate the fundamental
performance trade-offs between tracking and disturbance rejection. Relying on the multi-H∞ controller design presented
in [11], this paper presents a systematic and quantitative
analysis of these trade-offs for the overhead crane. In addition
to the valuable insight this yields in the underlying limits
of the performance of the control problem, it confirms the
applicability of the design approach.
The proposed control configuration is similar to the classical
two degrees-of-freedom (TDOF) control [15] in the sense
that it involves a two-input single-output controller. However,
in our control configuration both controller inputs comprise
a feedback signal, whereas in the classical TDOF control
the reference command constitutes one of the control inputs. In addition, the proposed design methodology differs
substantially from the classical TDOF control approach. The
latter designs the two controller components sequentially and
hence decouples disturbance rejection and reference tracking,
whereas our approach considers both swing angle disturbance
rejection and reference tracking simultaneously. Moreover,
the proposed multi-H∞ control design approach allows for a
systematic analysis of the trade-off between these performance
specifications.
Exploiting the SMILE interpolation does not guarantee
closed-loop stability under scheduling parameter variations.
In order to validate the GS controller there exist different
approaches to provide a stability certificate for the closedloop system with time-varying parameters (e.g. [16], [17]). In
this paper, the stability and performance of the GS controller
for the overhead crane is demonstrated by experiments with
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
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variable. To verify the effect of zero-order hold and possible
delays caused by the hardware and software configurations
(z)
, a frequency-domain
in the discrete-time model of XUc(z)
identification is performed. The input-output data used for
this identification is obtained by exciting the input u by a
multi-sine signal with frequency range [0.05, 5] Hz and period
20 sec [19]. The result is the following transfer function:
xc
θ
`
Xc (z)
0.0005
=
,
U (z)
z(z − 1)
xL
(a) Experimental test setup
Fig. 1.
-
dθ
ec
θ
Fig. 2.
which confirms the high bandwidth of the cart’s velocity
control loop since no additional dynamics originating from
the velocity controller is apparent in the frequency range of
excitation. There is also one additional delay (z −1 ).
In order to derive the relation between θ and xc , we use
Lagrange equations for a fixed cable length `, which yields
(b) Schematic diagram
The overhead crane test setup and its schematic diagram.
r +
K(`)
u
xc
G(`)
xL
`2 θ̈ + `x¨c cos(θ) + g` sin(θ) = 0 ,
θ
Control configuration used for the overhead crane.
a time-varying scheduling parameter. This experimental validation complements the time-domain and frequency-domain
simulations presented in [11].
This paper is organized as follows: Section II describes
the overhead crane system test setup and the corresponding
control configuration. The LTI control problem formulation
together with trade-off analysis are presented in Section III.
The resulting GS controller and experimental validations are
described and demonstrated in Section IV. Conclusions and
final remarks follow in Section V.
II. E XPERIMENTAL TEST SETUP
The overhead crane system consists of a cart to which a
load is attached by a cable. Figure 1(a) shows a picture of
the experimental setup, while Figure 1(b) depicts the physical
system model. The cart position, load position, swing angle
and cable length are indicated by xc , xL , θ and `, respectively.
The experimental setup is identified and controlled at a
sampling frequency of 100 Hz using SMF Ketels EBOX
(which uses EtherCAT) in combination with Orocos (Open
RObot COntrol Software) [18] on a Linux operating system.
The control input u to the system is the speed reference for
the cart’s velocity control loop which ranges between 0 and
m
]. The outputs xc , θ and ` are measured by encoders (and
1[ sec
hence known at each time instant), while the load position is
calculated as xL = xc + `θ. The length ` can vary between
0.35[m] and 0.73[m].
Given the high bandwidth of the cart’s velocity control
loop, we may assume:
Xc (s)
k
= ,
U (s)
s
(2)
(1)
where k [ v msec ] is a system constant relating the speed reference
m
input u [v] to the cart velocity ẋc [ sec
] and s is the Laplace
(3)
where g is the gravitational acceleration. For small angle θ, this
−s2
=
. This transfer function correrelation equals XΘ(s)
(s)
2
c
s `+g
sponds to an LTI second order system with undamped p
complex
g
1
conjugated poles with natural frequency fn (`) = 2π
` Hz.
Hence, both fn (`) and the high frequency gain of the system
decrease as ` increases.
Using the bilinear transformation, the discrete-time model
of XΘ(z)
equals
c (z)
Θ(z)
−b(`)(z − 1)2
= 2
,
Xc (z)
z + a1 (`)z + a2
(4)
1
where b(`), a1 (`), a2 can be written as b(`) = `+2.45×10
−4 ,
−4
−2`+4.9×10
a1 (`) = `+2.45×10−4 and a2 = 1.
According to (2) and (4), it is evident that the system
dynamics are LPV with ` being the scheduling parameter. It
is also worth mentioning that the hoisting system is velocity
controlled with a bandwidth that is at least eight times higher
than the resonance frequency of the system and hence, its
dynamics are neglected.
This system is controlled by feeding back (ec , θ) according
to the configuration outlined in Figure 2. The system G(`)
and K(`) represent the overhead crane and the controller as
a function of the cable length `. Input r is the reference
command and the disturbance input dθ imposes a swing angle
disturbance such that the system response to an impulse in dθ
corresponds to the autonomous response from an initial swing
angle θ0 6= 0 rad. Section III discusses the LTI control design
for this system.
III. LTI CONTROL
The primary goal of the controller K(`) is to make the
load follow the reference command r while rejecting the swing
angle disturbances. In addition, the control signal should stay
within the saturation limits of the actuator.
Superior load tracking requires a controller that does not
excite the system’s resonance and hence, has an anti-resonance
at the undamped natural eigenfrequency fn (`). However, such
a controller cannot compensate for load swinging due to any
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
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minimize
K∈S6 ,γe ,γd
γe + αγd
subject to kWe Hr,eL k∞ < γe
(5a)
(5b)
kHdθ ,θ k∞ < γd
(5c)
kWu Hr,u k∞ < 1,
(5d)
where S6 denotes the set of internally stabilizing controllers of
order six (full order controller). This problem is numerically
solved using the Lyapunov shaping paradigm presented in
[20]. It is well known that this method introduces a convex,
yet conservative solution approach for general multi-objective
control problems.
A. Trade-off analysis
To analyze the trade-off between load tracking and angle
disturbance rejection, a trade-off curve between γe and γd
is computed for the shortest cable length ` = 0.35[m]. This
is done by solving (5) using MATLAB in combination with
the SeDuMi solver [21], and the YALMIP interface [22] for
different values of α and plotting the optimal γe and γd values
as a function of each other. This result is shown in blackdashed line in Figure 3.
By increasing α, the trade-off curves are traced from left
to right, yielding better angle disturbance rejection, lower γd
value, at the cost of degraded load tracking, higher γe value.
The left-most point corresponds to the solution of (5) with
α = 0. The corresponding optimal controller is indicated by
K1 , and the optimal performance indices equal (γe,1 , γd,1 ) ≈
(0.32, ∞). The right-most point on the curve is obtained by
considering α = ∞, or equivalently, by replacing the objective
(5) by γd . The corresponding controller is indicated by K2 and
its performance indices equal (γe,2 , γd,2 ) = (10.62, 0.5).
6
actual value
optimal value
4
γd
angle disturbance, as this compensation requires a harmonic
control signal with frequency fn (`). Consequently, load tracking and angle disturbance rejection are conflicting performance
specifications in the controller design. To analyze this trade-off
using the multi-H∞ controller design, both specifications are
quantified in terms of a weighted closed-loop H∞ norm for
the shortest length ` = 0.35[m]. Load tracking is quantified
by γe = kWe Hr,eL k∞ where Hr,eL denotes the closedloop transfer function from r to eL = r − xL , and We =
0.0126/(z−0.999). This weight enforces Hr,eL to have a slope
of at least 20 [dB/dec] at low frequencies, and minimizing
γe corresponds to maximizing the bandwidth. Disturbance
rejection is also quantified by γd = kHdθ ,θ k∞ , such that
minimizing γd corresponds to maximizing the damping of
fn (`) in Hdθ ,θ : the closed-loop transfer function from dθ to
θ. As a third specification in the controller design problem, a
constraint on the actuator effort is added: kWu Hr,u k∞ < 1,
where Hr,u denotes the closed-loop transfer function from r to
u and Wu = (z−0.999)/(20z−9.4) to bound the derivative of
u, which equals the cart acceleration. This weighting function
is iteratively tuned to avoid input saturation for displacement
steps of 0.1[m].
To conclude, analyzing the trade-off between load tracking
and angle disturbance rejection under bounded actuator effort
amounts to solving the following control problem for various
values of the dimensionless weight α:
2
0
0.4
0.6
0.8
1
1.2
1.4
γe
Fig. 3. Trade-off curves of tracking performance and disturbance rejection
for shortest cable length `= 0.35 m. The square-blue point is the tradeoff point using K3 which provides a good compromise between conflicting
performances. The axes range is restricted for visual improvements.
The steep left part of the trade-off curve indicates that
starting from K1 , γd can be decreased significantly with only
a small increase in γe . The shallow right part, on the other
hand, means that a controller can achieve significantly better
load tracking performance than K2 while having only slightly
worse angle disturbance rejection. A good engineering choice
would therefore be a controller that corresponds to a point on
the trade-off curve with a moderate slope part, as for instance
K3 , which is indicated by the blue square in Figure 3. In order
to show this moderate slope better, Figure 3 depicts a limited
region of the entire trade-off curve and hence the controllers
K1 and K2 are not visible.
To analyze the conservatism caused by the Lyapunov shaping paradigm used to solve (5), the gray dotted line in
Figure 3 shows the true values of γe = kWe Hr,eL k∞ and
γd = kHdθ ,θ k∞ achieved by the optimized controllers. Due
to the conservatism, the optimized γe , γd values, obtained
by solving (5) using Lyapunov shaping paradigm, provide
only an upper bound to the true performance indices achieved
by the optimized controller. The distance between the curves
is a measure for the conservatism of the Lyapunov shaping
paradigm.
In order to support the findings of the previous paragraph,
the controllers K1 , K2 and K3 are further compared and
implemented on the verified model of the overhead crane
((2) and (4)) for the shortest length ` = 0.35[m]. Figure 4
compares closed-loop frequency response functions (FRFs)
of Hr,eL , Hr,u and Hdθ ,θ , while Figure 5 shows their timedomain responses. The results for K1 , which yields the best
load tracking, are shown in dashed green. The dotted purple
lines correspond to K2 , which yields the best disturbance
rejection, while the solid blue lines relate to K3 , which yields
a good trade-off between load tracking and angle disturbance
rejection. In Figure 5 the top plot shows the response to a step
input of 10 cm in r and the lower plot depicts an initial angle
deviation.
Figure 4 confirms that K1 yields the highest bandwidth
for Hr,eL . To achieve this superior tracking performance K1
must not excite the system’s resonance, due to which it cannot
counteract the swinging initiated by an initial angle deviation.
Consequently, in the FRF of Hdθ ,θ the system’s undamped
resonance prevails. Figure 5 confirms the superior tracking
and poor disturbance rejection performance of K1 .
The controller K2 yields superior disturbance rejection
at the cost of a very poor tracking. This is confirmed by
5
6
0
4
γd
|Hr,eL | [dB]
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
−20
2
−40
0
|Hdθ ,θ | [dB]
|Hr,u | [dB]
40
0.8
1
20
0
1.2
1.4
Fig. 6. Trade-off curves of tracking performance and disturbance rejection for
different cable lengths ` = [0.35, 0.55, 0.63, 0.73][m] ranging from dotted
blue to solid red line. The square point is the trade-off point which provides
a good compromise between conflicting performances. The axes range is
restricted for visual improvements.
−20
40
20
0
−20
10−1
100
101
Freq [Hz]
Fig. 4. Closed-loop Bode diagrams using the controller K1 (dashed green),
K2 (dotted purple) and K3 (solid blue).
Cart pos [m]
0.6
γe
10−2
decreases with longer cable lengths. Therefore, the bandwidth
decreases for larger ` and the trade-off plot of longer cable
length shifts higher. This means that for the same disturbance
rejection performance the tracking gets worse for local systems
corresponding to longer cable length. The local LTI controllers
selected for interpolation into a GS controller using SMILE
technique are indicated with squares in Figure 6. The results
of this interpolation is discussed in the next section.
0.1
IV. I NTERPOLATING GS CONTROLLER
This section evaluates the GS controller on the overhead
crane by interpolating the four selected LTI local controllers
using the SMILE interpolation technique [12]. This technique
formulates the interpolation as a linear least squares problem
and calculates the state-space coefficients of the parameter
dependent model as a function of the varying parameter.
The interpolation result is a polynomially parameter dependent controller of degree two. In other words, any controller
in the range of variation [0.35, 0.73][m] is calculated as:
0.05
0
1
Angle [rad]
0.4
·10−2
0
−1
0
1
2
3
4
Time [sec]
K(`) =
A(`) B(`)
C(`) D(`)
= K0 + `K1 + `2 K2 .
(6)
Fig. 5. Reference step response (top figure) and Swing angle disturbance
response (bottom figure) using controller K1 (dashed green), K2 (dotted
purple) and K3 (solid blue).
where Ki , i = 0, 1, 2 are matrix coefficients of the GS
controller as the following:
Ai Bi
Ki =
, i = 0, 1, 2
(7)
Ci Di
frequency- and time-domain responses. Controller K3 , on
the other hand, yields a good compromise between the two
performance aspects. Compared to K1 it yields a slightly lower
bandwidth in Hr,eL , while providing much more damping to
the resonance in Hdθ ,θ . Compared to K2 , K3 gives a slightly
higher kHdθ ,θ k∞ , while providing a significantly faster time
response as depicted in Figure 5.
Following the same methodology, we can design local
LTI controllers for identified local models corresponding to
different cable length values. Figure 6 shows the tradeoff figure for actual γ values for cable lengths of ` =
[0.35, 0.55, 0.63, 0.73][m] ranging from the most dotted blue
to solid red. The curves of larger ` have higher γ values which
is a consequence of the bound on the actuator effort imposed
in the optimization problem (5). The bound is only active at
high frequencies and the high frequency gain of system (4)
The validation in this section comprises three parts. In the
first two parts the cable length is assumed to be fixed while in
the third part the validation is done for varying cable length.
The first part is to compare the FRFs of the GS controller
evaluated at different cable length values with the local LTI
controllers in order to validate the interpolation accuracy. In
the second part, the corresponding closed-loop systems using
the GS controller and the local LTI controllers are compared.
This is done to verify the closed-loop performance using the
GS controller. The third and final part experimentally validates
the GS controller for time-varying parameters.
In Figure 7 the calculated GS controller is compared to
the local LTI controllers. The gray lines show the magnitude
of the frequency response of the GS controller evaluated at
different ` values, while the black lines changing from blue
to red are the four local LTI controllers. Although a small
10
0
0.8
0.6
0.4
ℓ [m]
−2
10
−1
10
0
10
Freq. [Hz]
1
10
(a) Channel 1: Kec ,u
0.5
0.1
0.45
0.4
0.35
0.3
12
10
8
0.05
0.1
0
4
0.4
0.05
0.5
0.6
0.7
` [m]
10
0.8
0.6
0.4
ℓ [m]
−2
10
−1
10
0
10
Freq. [Hz]
1
10
(b) Channel 2: Kθ,u
Fig. 7. Local LTI controllers (black) and interpolating GS controller of
degree two evaluated for 19 equidistant values of ` (gray).
mismatch is visible on the first channel of the controller at
high frequencies, the error is not important as it only appears
at frequencies much higher than the closed-loop bandwidth.
Therefore comparable performance can be expected by using
the GS controller.
Figure 8 illustrates the effect of interpolation on the closedloop bandwidth and the norm kHdθ ,θ k∞ as a function of `.
Closed-loop systems corresponding to the local LTI controllers
and the GS controller evaluated at different ` values, are
indicated with black-squares and green-star respectively. As
it is depicted, the bandwidth decreases as the cable length
increases and comparison shows that the gain-scheduled controller has a negligible lower bandwidth (less than 2%). Figure
8 also shows that the norm value kHdθ ,θ k∞ with the GS
controller is slightly lower (6% at the most). The difference
between the gain-scheduled and local LTI controller is due to
the interpolation error and it is insignificant since the gainscheduled controller gives satisfactory performance.
In order to validate the GS controller with time-varying
parameters, a set of experiments with varying cable length
are performed. In this paper three representative results are
shown: The step response with a length value varying from
` = 0.73[m] to ` = 0.35[m] at three different constant rates of
m
]. Figure 9 shows the step responses and
0.2, 0.1 and 0.05 [ sec
2
0
4
0.1
0
0.05
0.1
2
0.8
Control Signal [v]
20
0
4
0.05
0.6
0
2
0
0.8
0
0.4
4
0.6 0
0.8
` [m]
30
Load position [m]
Fig. 8.
Closed-loop bandwidth (top) and kHdθ ,θ k∞ as a function of `
(bottom). Local LTI controllers (black squares) are compared to the gainscheduled controller (solid green) evaluated for 19 equidistant values of `.
Control Signal
[v]
` [m]
Mag. [dB]
40
dθ ,θ ∞
20
0.6
2
4
0.4
0
2
2
0.4
4
6
8
Time [s
Time [sec]
Fig. 9. Position of the load in presence of parameter variation. Fast parameter
variation (solid red) compared to slower parameter variations (dotted green
and dashed blue). The reference command is shown in black dash-dotted.
0
0
2
4
Time [s
cable length for fast to slow variation. The fastest variation
m
shown in solid-red line is set to 0.2[ sec
] such that the load
is at the desired position when the step response is settled.
The length change is applied at t = 0.1[sec] right after the
reference command is set to 0.1[m] and as it is shown there
is barely any difference in performance at fast or slow length
variations.
0.8
` [m]
Mag. [dB]
30
6
kH
k [dB]
Bandwidth [Hz]
Load[v]
position
[m]
Load position
[m]
Load
Control
position Signal
[m] Control
[v]
Signal
` [m]
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
0.6
0.4
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, JUNE 2013
V. C ONCLUSION
This paper extends the practical interpolated GS controller
design approach for LPV systems presented in [11], by thoroughly investigating the approach and experimentally confirming the results on an overhead crane system with a varying cable length. In the first step, LTI control design is performed by
splitting the classical H∞ control problem into H∞ constraints
and objectives on selected closed-loop subsystems. This multiH∞ controller design method allows us to analyze the natural
trade-off between reference tracking and disturbance rejection
present in the overhead crane system. The second step, is using
a recently developed interpolation technique (SMILE) for LPV
modeling to derive gain-scheduled controllers from local LTI
controllers. The experimental results on the system reveal the
value of this approach in presence of varying parameters.
ACKNOWLEDGEMENTS
This work benefits from KU Leuven-BOF PFV/10/002
Center-of-Excellence Optimization in Engineering (OPTEC),
the Belgian Programme on Interuniversity Attraction Poles,
initiated by the Belgian Federal Science Policy Office
(DYSCO), European research project EMBOCON FP7-ICT2009-4 248940, KU Leuven’s Concerted Research Action
GOA/10/11, and G.0377.09 of the Research Foundation Flanders (FWO Vlaanderen). Goele Pipeleers is a Postdoctoral
Fellow of the Research Foundation - Flanders.
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Keivan Zavari received his M.Sc. degree (with
high honors) in control engineering from Tarbiat
Modares University, Tehran, Iran, in 2009. He is
currently working toward a Ph.D. in the Department
of Mechanical Engineering, Division Production Engineering, Machine Design and Automation (PMA)
of the KU Leuven. His current research is ”Design
of robust feedback controllers for linear parameter
varying (LPV) systems”. His research interests include control theory, convex optimization, and their
applications to mechatronic systems.
Goele Pipeleers received her M.Sc. and Ph.D. degree in mechanical engineering from the KU Leuven, Belgium, in 2004 and 2009, respectively. Currently she is a Post-doctoral Fellow of the Research
Foundation-Flanders affiliated with the KU Leuven,
Department of Mechanical Engineering, Division
of Production Engineering, Machine Design and
Automation (PMA). Her research interests include
convex optimization, optimal and robust control, and
their applications to mechatronic systems.
Jan Swevers received his M.Sc. degree in electrical engineering and the Ph.D. degree in mechanical engineering from the Katholieke Universiteit Leuven (KU Leuven), Belgium, in 1986 and
1992, respectively. He is since 1995 a professor in the Department of Mechanical Engineering, Division Production Engineering, Machine Design and Automation (PMA), of KU Leuven. His
research interests include modeling, identification,
control and optimization of mechatronics systems.
http://www.mech.kuleuven.be/meco